Overview and Motivation:

Some of our team members are loyal fans of the popular free online card game, Hearthstone: Heroes of Warcraft, which was released worldwide by Blizzard on 2014 with more than 40 million registered Hearthstone accounts by November 2015.

The main element of the game Hearthstone are cards, which consist of a list of features including cost, attack (number of damages can be made to the opponent per turn),health (number of damages that can bear before being destroyed) and other special abilities. Here is an example of the card:

Before every game starts, each of the two players will choose 1 hero mode among the 9 and then select 30 different cards over 700 cards to build his/her own deck depending on the mode. Each turn, the player will draw one card randomly from the 30 cards and one more mana crystal (money). The player can choose the cards to use among all those in hand that cost up to the mana crystals he/she has by that turn. The game ends when one player is attacked to death (lose all 30 units of health) or he/she concedes, and the other player will win.

Therefore, the initial building of the 30 cards, as well as the choices of cards to use during the game will directly influence the results of the game. This motivated us:

Initial Questions:

1. What are the “true” values of individual cards? Are there any properties the Blizard company used to assign values (cost) of these cards? Is there any card undervalued/overvalued by the company?_

2. What is the balance between low cost cards and high cost cards?_

3. Are there any “core” combination of cards?_

4. Are we able to build a powerful deck (30 cards) for some heros?_

5. Test the deck we built (optional)_ * We can test our model by simulating games using the deck and strategy we developed, and calculate its percentage of winning. ## Related Work:

library

Here are the libraries we have used in our project.

library(rjson)
library(dplyr)
library(tidyr)
library(knitr)
library(readr)
library(stringr)
library(ggplot2)
library(gridExtra)
library(graphics)
library(grid)
library(ggrepel)
library(scales)
library(cowplot)
library(rvest)
library(XML)
library(vegan)
library(RColorBrewer)
library(gplots) 
library(devtools)
library(reshape)
library(dendextend)
library(reshape2)
library(VGAM)

Data:

We have two types of data: 1) basic card information (attack/health/cost/description of cards) and 2) frequently used decks from top players.

## Data wrangling from json to RData:
json_file = "cards2.txt"
data <- fromJSON(file = json_file)
card_category = names(data)

not_empty = which(sapply(1:length(data), function(i){length(data[[i]])})>0)

card_category = card_category[not_empty]

data = lapply(not_empty, function(i){data[[i]]})
data1 = lapply(1:length(data), function(k) {lapply(data[[k]],
                                                   function(i) {lapply(i, function(j){
                                                     j = ifelse(is.null(j),NA,j)})})})

col_names = lapply(1:length(data1),
                   function(k) {
                     lapply(1:length(data1[[k]]),
                            function(i) {names(data1[[k]][[i]])})})

data2 = lapply(1:length(data1), 
           function(k) {
             lapply(1:length(data1[[k]]),
                    function(i) {
                      matrix(unlist(data1[[k]][[i]]), 
                             ncol = length(data1[[k]][[i]]), 
                             byrow = T)})})

for(k in 1:length(data2)){
  colnames(data2[[k]][[1]]) = col_names[[k]][[1]]
  data2[[k]][[1]] = data.frame(data2[[k]][[1]])
  for(i in 2:length(data2[[k]])){
    colnames(data2[[k]][[i]]) = col_names[[k]][[i]]
    data2[[k]][[i]] = data.frame(data2[[k]][[i]])
    data2[[k]][[i]] = bind_rows(data2[[k]][[i-1]],data2[[k]][[i]])
  }
  assign(card_category[k], tbl_df(data2[[k]][[length(data2[[k]])]]))
}

final_data = get(card_category[1])
for (i in 2:length(data2)){
  final_data = bind_rows(final_data, get(card_category[i]))
}
# write.table(final_data, file = "final_data.csv", sep = "\t")
# save(final_data, file = "final_data.RData")

Data wrangling of card descriptions: This part is aimed for detailed classification of minion card descriptions (other than the mechanics they are currently classified as).

  1. For some minions that have more than one mechanics (e.g. Taunt that has deathrattle), they are only classified as one of their mechanics in the Hearthstone dataset. The following wrangling aims to classify them with all the mechanics they have with dummy variables (having certain feature = 1 vs. without certain feature = 0).
load("minions_text.RData")
minions_text = tbl_df(minions_text) %>%
  select(-cardId, -flavor, -type, -artist, -collectible, -howToGet, -howToGetGold, -img, -imgGold, -locale, -race, -faction, -elite) %>%
  mutate(playerClass = ifelse(is.na(playerClass), "All", as.character(playerClass)))

minions_text = minions_text %>% 
  mutate(text = as.character(text)) %>%
  mutate(text = gsub("<b>", "", text)) %>%
  mutate(text = gsub("</b>", "", text)) %>%
  mutate(text = gsub("\xa1\xaf", "'", text)) %>%
  mutate(text = ifelse(is.na(text), "None", text)) 

minions_text = minions_text %>%
  mutate(AdjacentBuff= ifelse(text %in% minions_text$text[grep("AdjacentBuff",minions_text$text)], 1, AdjacentBuff))%>% 
   mutate(Aura= ifelse(text %in% minions_text$text[grep("Aura",minions_text$text)], 1, 0))%>% 
   mutate(Battlecry = ifelse(text %in% minions_text$text[grep("Battlecry",minions_text$text)], 1, Battlecry))%>%
  mutate(Charge= ifelse(text %in% minions_text$text[grep("Charge",minions_text$text)], 1, Charge))%>%
 mutate(Combo = ifelse(text %in% minions_text$text[grep("Combo",minions_text$text)], 1, Combo))%>%
  mutate(Deathrattle = ifelse(text %in% minions_text$text[grep("Deathrattle",minions_text$text)], 1, Deathrattle))%>%
  mutate(Divine_Shield = ifelse(text %in% minions_text$text[grep("Divine_Shield",minions_text$text)], 1, Divine_Shield))%>%
  mutate(Enrage = ifelse(text %in% minions_text$text[grep("Enrage",minions_text$text)], 1, Enrage))%>%
  mutate(Inspire = ifelse(text %in% minions_text$text[grep("Inspire",minions_text$text)], 1, Inspire))%>%
  mutate(Overload= ifelse(text %in% minions_text$text[grep("Overload",minions_text$text)], 1, Overload))%>%
  mutate(Poisonous = ifelse(text %in% minions_text$text[grep("Poisonous",minions_text$text)], 1, Poisonous))%>%
  mutate(Windfury = ifelse(text %in% minions_text$text[grep("Windfury",minions_text$text)], 1, Windfury))
  1. The beauty of Hearthstone (and the most difficult part for quantitative analysis) is that almost every minion has its unique feature that are described in text on the card. Therefore, it’s hard to “value” a card without taking these descriptions into account while on the other hand, texts them selves are difficult to be simply quantified. The following wrangling aims to identify certain verbs (deal, restore, etc.) and nouns (attacks, healths, etc.) frequently used in the card description and tried to classifiy cards with more features that were not classified by their mechanics.
minions_text = minions_text %>%
  mutate(Choice = ifelse(text %in% minions_text$text[grep("; or",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Conditional = ifelse(text %in% minions_text$text[grep("if",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Conditional = ifelse(text %in% minions_text$text[grep("whenever",minions_text$text, ignore.case = T)], 1, Conditional)) %>% 
  mutate(Conditional = ifelse(text %in% minions_text$text[grep(",",minions_text$text, ignore.case = T)], 1, Conditional)) %>% 
  mutate(Add = ifelse(text %in% minions_text$text[grep("add",minions_text$text, ignore.case = T)], 1, 0)) %>%
  mutate(Cast = ifelse(text %in% minions_text$text[grep("cast",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Deal = ifelse(text %in% minions_text$text[grep("Deal",minions_text$text, ignore.case = T)], 1, 0)) %>%
  mutate(Destroy = ifelse(text %in% minions_text$text[grep("destroy",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Discover = ifelse(text %in% minions_text$text[grep("discover",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Draw = ifelse(text %in% minions_text$text[grep("draw",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Discard = ifelse(text %in% minions_text$text[grep("discard",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Freeze = ifelse(text %in% minions_text$text[grep("freeze",minions_text$text, ignore.case = T)], 1, Freeze)) %>% 
  mutate(Gain = ifelse(text %in% minions_text$text[grep("gain",minions_text$text, ignore.case = T)], 1, 0)) %>%
  mutate(Give = ifelse(text %in% minions_text$text[grep("give",minions_text$text, ignore.case = T)],1,0)) %>%
  mutate(Reduce = ifelse(text %in% minions_text$text[grep("reduce",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Remove = ifelse(text %in% minions_text$text[grep("remove",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Restore = ifelse(text %in% minions_text$text[grep("restore",minions_text$text, ignore.case = T)], 1, 0))%>%
  mutate(Reveal = ifelse(text %in% minions_text$text[grep("reveal",minions_text$text, ignore.case = T)],1,0)) %>%
  mutate(Silence = ifelse(text %in% minions_text$text[grep("silence",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Summon = ifelse(text %in% minions_text$text[grep("summon",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Trigger = ifelse(text %in% minions_text$text[grep("trigger",minions_text$text, ignore.case = T)],1,0)) %>%
  mutate(Number_within = ifelse(text %in% minions_text$text[grep("+[0-9]", minions_text$text)],1,0))%>%
  mutate(Attack = ifelse(text %in% minions_text$text[grep("attack",minions_text$text, ignore.case = T)], 1, 0))%>%
  mutate(Health = ifelse(text %in% minions_text$text[grep("health",minions_text$text, ignore.case = T)], 1, 0))%>%
  mutate(Damage = ifelse(text %in% minions_text$text[grep("damage",minions_text$text, ignore.case = T)], 1, 0)) %>%
  mutate(Cant = ifelse(text %in% minions_text$text[grep("can't",minions_text$text, ignore.case = T)], 1, 0)) %>%
  mutate(Nothing = ifelse(text == "None", 1, 0))

colnames(minions_text)

save(minions_text, file = "minions_text.RData")

Exploratory Analysis

theme_set(theme_bw(base_size = 16))
#setwd("/Users/Scarlett/Desktop/final")
load("minions_text.RData")
data<-minions_text

distribution of Cost

#remove costs that are "12" and "20" for these two cards are very special
Cost<-data%>%dplyr::arrange(cost)
Cost<-unique(data%>%filter(cost<=10)%>%group_by(cost)%>%mutate(n=n())%>%ungroup()%>%select(cost,n))
# 7 stands for higher than 7
Cost1<-Cost%>%filter(cost<7)                       
Cost2<-c(7,61)
Cost<-rbind(Cost1,Cost2)

Cost<-Cost%>%mutate(pos=cumsum(n)-n/2)
p<-Cost%>%ggplot(aes(x=1,y=n,fill=factor(cost)))
p+geom_bar(stat="identity",width=1)+geom_text(aes(x=1.6,y=pos,label = percent(n/sum(n))))+coord_polar(theta="y")+ xlab('')+ylab('')+theme(axis.text=element_blank(),axis.ticks=element_blank(),panel.grid=element_blank())+ggtitle("Pie Chart of Card Cost")

#histogram
ggplot(data, aes(factor(cost)))+ geom_bar()+scale_fill_brewer()+ggtitle("Histogram for cost")+xlab("cost")

Conclusion: cards with cost “2”,“3”,“4” out of the 11 possible costs occupying around 54% in total are most common in the deck

distribution of attack

Attack<-data%>%arrange(attack)
Attack<-unique(Attack%>%group_by(attack)%>%mutate(n=n())%>%ungroup()%>%select(attack,n))
#histogram
ggplot(data, aes(factor(attack)))+ geom_bar()+scale_fill_brewer()+ggtitle("Histogram for attack")+xlab("attack")

distribution of health

Health<-data%>%arrange(health)
Health<-unique(Health%>%group_by(health)%>%mutate(n=n())%>%ungroup()%>%select(health,n))
#histogram
ggplot(data, aes(factor(health)))+ geom_bar()+scale_fill_brewer()+ggtitle("Histogram for health")+xlab("health")

distribution of mechanics

Mechanics<-data%>%arrange(mechanics)
Mechanics<-unique(Mechanics%>%group_by(mechanics)%>%mutate(n=n())%>%ungroup()%>%select(mechanics,n))
#histogram
ggplot(data, aes(factor(mechanics)))+ geom_bar()+scale_fill_brewer()+ggtitle("Histogram for mechanics")+xlab("mechanics")

*distribution of each mechanics

qplot(factor(data$mechanics),data$attack,geom="boxplot")+scale_fill_brewer()+theme(axis.text.x = element_text(angle = 45, hjust = 1,size=10))+ggtitle("Boxplots for cost in mechanics group")+xlab("Rarity")+ylab("cost")

qplot(factor(data$mechanics),data$attack,geom="boxplot")+scale_fill_brewer()+theme(axis.text.x = element_text(angle = 45, hjust = 1,size=10))+ggtitle("Boxplots for health in mechanics group")+xlab("Rarity")+ylab("health")

qplot(factor(data$mechanics),data$attack,geom="boxplot")+scale_fill_brewer()+theme(axis.text.x = element_text(angle = 45, hjust = 1,size=10))+ggtitle("Boxplots for Attack in mechanics group")+xlab("Rarity")+ylab("attack")

distribution of cardSet

cs<-data%>%arrange(cardSet)
cs<-unique(cs%>%group_by(cardSet)%>%mutate(n=n())%>%ungroup()%>%select(cardSet,n))
#pie chart
cs<-cs%>%mutate(pos=cumsum(n)-n/2)
p<-cs%>%ggplot(aes(x=1,y=n,fill=factor(cardSet)))
p+geom_bar(stat="identity",width=1)+geom_text(aes(x=1.6,y=pos,label = percent(n/sum(n))))+coord_polar(theta="y")+ xlab('')+ylab('')+theme(axis.text=element_blank(),axis.ticks=element_blank(),panel.grid=element_blank())+ggtitle("Pie Chart of cardSet")

#histogram
qplot(data$cardSet,xlab="cardSet",main="Histogram for CardSet distribution")+theme(axis.text.x = element_text(angle = 45, hjust = 1))

*distribution of each CardSet

qplot(factor(data$cardSet),data$attack,geom="boxplot")+scale_fill_brewer()+theme(axis.text.x = element_text(angle = 45, hjust = 1,size=10))+ggtitle("Boxplots for cost in CardSet group")+xlab("Rarity")+ylab("cost")

qplot(factor(data$cardSet),data$attack,geom="boxplot")+scale_fill_brewer()+theme(axis.text.x = element_text(angle = 45, hjust = 1,size=10))+ggtitle("Boxplots for health in CardSet group")+xlab("Rarity")+ylab("health")

qplot(factor(data$cardSet),data$attack,geom="boxplot")+scale_fill_brewer()+theme(axis.text.x = element_text(angle = 45, hjust = 1,size=10))+ggtitle("Boxplots for Attack in CardSet group")+xlab("Rarity")+ylab("attack")

distribution of rarity

rr<-unique(data%>%group_by(rarity)%>%mutate(n=n())%>%ungroup()%>%select(rarity,n))
#pie chart
rr<-rr%>%mutate(pos=cumsum(n)-n/2)
p<-rr%>%ggplot(aes(x=1,y=n,fill=factor(rarity)))
p+geom_bar(stat="identity",width=1)+geom_text(aes(x=1.6,y=pos,label = percent(n/sum(n))))+coord_polar(theta="y")+ xlab('')+ylab('')+theme(axis.text=element_blank(),axis.ticks=element_blank(),panel.grid=element_blank())+ggtitle("Pie Chart of rarity")

#histogram
qplot(data$rarity,xlab="rarity",main="Histogram for rarity")+theme(axis.text.x = element_text(angle = 45, hjust = 1))

*distribution for each rarity

qplot(factor(data$rarity),data$attack,geom="boxplot")+scale_fill_brewer()+theme(axis.text.x = element_text(angle = 45, hjust = 1,size=10))+ggtitle("Boxplots for cost in Rarity group")+xlab("Rarity")+ylab("cost")

qplot(factor(data$rarity),data$attack,geom="boxplot")+scale_fill_brewer()+theme(axis.text.x = element_text(angle = 45, hjust = 1,size=10))+ggtitle("Boxplots for health in Rarity group")+xlab("Rarity")+ylab("health")

qplot(factor(data$rarity),data$attack,geom="boxplot")+scale_fill_brewer()+theme(axis.text.x = element_text(angle = 45, hjust = 1,size=10))+ggtitle("Boxplots for Attack in Rarity group")+xlab("Rarity")+ylab("attack")

Final Analysis:

Card Value Analysis

1. What are the “true” values of individual cards? Are there any properties the Blizard company used to assign values (cost) of these cards? Is there any card undervalued/overvalued by the company?

load("minions_text.RData")
## cost vs attack+health:
minions_text %>% ggplot(aes(cost)) + stat_bin(aes(y = ..count..), bins = 50 , position='dodge')

minions_text %>% mutate(attplusheal = attack+health) %>% ggplot(aes(attplusheal)) + stat_bin(aes(y = ..count..), bins = 50 , position='dodge')

From the above plots, we can found similar distributions between the cost and the sum of attach and health, where the distributions are right-skewed. Also, there seems to be some outliers that are very different from other cards.

minions_text %>%
  filter(cost > 10) %>%
  select(name, cost, attack, health, mechanics, playerClass)
## Source: local data frame [3 x 6]
## 
##              name  cost attack health mechanics playerClass
##            (fctr) (int)  (int)  (int)     (chr)       (chr)
## 1  Mountain Giant    12      8      8    Normal         All
## 2    Molten Giant    20      8      8    Normal         All
## 3 Clockwork Giant    12      8      8    Normal         All

It might be a good idea to filter out these cards.

highcost_card = minions_text %>% 
  mutate(attplusheal = attack+health) %>% filter(cost > 10) %>% 
  mutate(cost1 = 7) %>%
  select(cardId, name, cost,cost1,mechanics,Charge, Overload, attack, health)
  

minions_text = minions_text %>% mutate(attplusheal = attack+health) %>% filter(cost <= 10)

## cost vs attack+health:
minions_text %>% mutate(attplusheal = attack+health) %>% 
  group_by(attplusheal) %>%
  summarize(cost = mean(cost)) %>%
  ggplot(aes(attplusheal, cost)) + geom_point()

We can see from the above graph that higher attplusheal value (attack+health) is associated with higher mean cost.

In Hearthstone, the cost of cards is usually categorized into 0 ~ 6 and 7+. Here, we wrangled the card costs into these 8 categories and also separate them by cardSet:

## All:
minions_text = minions_text %>% 
  mutate(cost1 = ifelse(cost >= 7, 7, cost))
minions_text %>% ggplot(aes(cost1)) + geom_histogram()
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

minions_text %>% mutate(attplusheal = attack+health) %>% 
  group_by(cost1, attplusheal) %>% summarize(count = n()) %>%
  ggplot(aes(attplusheal, cost1, col = factor(floor(count/10)*10))) + geom_point()

## by cardSet:
minions_text %>% ggplot(aes(cost1, group = cardSet, fill = cardSet)) + stat_bin(aes(y = ..count..), bins = 20 , position='dodge')

## by attack, cost, health:
minions_bars = minions_text %>% gather(key, value, cost, attack, health)
minions_bars %>% ggplot(aes(value, group = key, fill = key)) + stat_bin(aes(y = ..count..), bins = 40, position='dodge')

## by cardSet:
## Cost:
minions_text %>% ggplot(aes(cost, group = cardSet, fill = cardSet)) + stat_bin(aes(y = ..count..), bins = 40 , position='dodge')

## Attack:
minions_text %>% ggplot(aes(attack, group = cardSet, fill = cardSet)) + stat_bin(aes(y = ..count..), bins = 40 , position='dodge')

## Health:
minions_text %>% ggplot(aes(health, group = cardSet, fill = cardSet)) + stat_bin(aes(y = ..count..), bins = 40 , position='dodge')

Since the outcome variable (Y) in our analysis is the costs of cards, which are normally integer from 0 to 7+ (all values greater than 7 are considered in the group of 7+), we adopted a model that consider ordinal polytomous outcome – cumulative logits model. Since the features’ effects (attack, cost, special abilities, etc.) should be the similar in cards with different costs, we also assumed proportional odds of these features across different cost groups. And we ended up with 7 outcome groups (cost value: 1 to 7+), we excluded cards that cost 0 mana since 1) they are usually cards that do not cost players to play and 2) the nature of these 0 cost cards are quite different from normal minion cards. In general, the cumulative logits model is in format shown below, where X is the covariate matrix, and \(\beta\) is the coefficient matrix:

\[\mbox{logit(Pr}{(Y \leq k | X_i = x_i))} = \beta_{k0} + \sum \beta_{ki}*x_i\]

Using this cumulative logits model, we are able to estimate the probability of a card being classified in each cost group (p1 to p7), and then by conditioning on the features of a card, we are able to assign a value of that card with the maximum probability among p1 to p7 (the most likely cost of a card based on its features).

Since one of our assumption that the cost of a card is proportional to the damage it can lead to, we first considered a univariate model which include attack as the only covariate:

## X: attack
## Y: cost

minions_text1 = minions_text %>% 
  mutate(mechanics1 = ifelse(mechanics %in% c("Charge","Divine Shield", "Overload", "Taunt", "Stealth", "Windfury"), mechanics, "A")) %>%
  filter(cost != 0) %>%
  arrange(cost) %>%
  mutate(Y1 = ifelse(cost == 1, 1, 0)) %>%
  mutate(Y2 = ifelse(cost == 2, 1, 0)) %>%
  mutate(Y3 = ifelse(cost == 3, 1, 0)) %>%
  mutate(Y4 = ifelse(cost == 4, 1, 0)) %>%
  mutate(Y5 = ifelse(cost == 5, 1, 0)) %>%
  mutate(Y6 = ifelse(cost == 6, 1, 0)) %>%
  mutate(Y7 = ifelse(cost >= 7, 1, 0)) 

set.seed(1001)
n_test <- round(nrow(minions_text1) / 10)
test_indices <- sample(1:nrow(minions_text1), n_test, replace=FALSE)
test <- minions_text1[test_indices,]
train <- minions_text1[-test_indices,]

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ attack, cumulative(parallel = T, reverse = F), data = train)
# summary(fitCL)

for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}

# To estimate the cost of cards based on attack:
test1 = test %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*attack)/(1+exp(coef1[1,]+coef1[2,]*attack)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*attack)/(1+exp(coef2[1,]+coef2[2,]*attack))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*attack)/(1+exp(coef3[1,]+coef3[2,]*attack))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*attack)/(1+exp(coef4[1,]+coef4[2,]*attack))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*attack)/(1+exp(coef5[1,]+coef5[2,]*attack))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*attack)/(1+exp(coef6[1,]+coef6[2,]*attack))) - p1 - p2 - p3 - p4 - p5) %>% mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

test1 = test %>% left_join(test1, by = "cardId")
RMSE <- function(true_ratings, predicted_ratings){
    sqrt(mean((true_ratings - predicted_ratings)^2))
}
model1 = RMSE(test1$cost1, test1$value)
rmse_results = data_frame(method = "X: attack", RMSE = model1)

Since the cost of a card can also be influenced by the time it can survive on the stage, we also included some potential effect of health by summing up both attack and health (attack+health ) as a univariate:

## X: attplusheal
## Y: cost

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ attplusheal, cumulative(parallel = T, reverse = F), data = train)
# summary(fitCL)

for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}

# To estimate the cost of cards based on attack plus health:
test2 = test %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*attplusheal)/(1+exp(coef1[1,]+coef1[2,]*attplusheal)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*attplusheal)/(1+exp(coef2[1,]+coef2[2,]*attplusheal))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*attplusheal)/(1+exp(coef3[1,]+coef3[2,]*attplusheal))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*attplusheal)/(1+exp(coef4[1,]+coef4[2,]*attplusheal))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*attplusheal)/(1+exp(coef5[1,]+coef5[2,]*attplusheal))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*attplusheal)/(1+exp(coef6[1,]+coef6[2,]*attplusheal))) - p1 - p2 - p3 - p4 - p5) %>% 
  mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

test2 = test %>% left_join(test2, by = "cardId")
model2 = RMSE(test2$cost1, test2$value)
rmse_results =  bind_rows(rmse_results, data_frame(method = "X: attplusheal", RMSE = model2))

It seemed like the univariate attack+health worked well in the model, as we testing the model in our testing set, the RMSE decreased. We also considered a model which include attack and health separately:

## X: attack, health
## Y: cost

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack, cumulative(parallel = T, reverse = F), data = train)
# summary(fitCL)

for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}


# To estimate the cost of cards based on attack and health:
test3 = test %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack)/(1+exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack)/(1+exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack)/(1+exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack)/(1+exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack)/(1+exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack)/(1+exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack))) - p1 - p2 - p3 - p4 - p5) %>% mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

test3 = test %>% left_join(test3, by = "cardId")
model3 = RMSE(test3$cost1, test3$value)
rmse_results = bind_rows(rmse_results,data_frame(method="X: attack, health",  
                                     RMSE = model3))
rmse_results
## Source: local data frame [3 x 2]
## 
##              method      RMSE
##               (chr)     (dbl)
## 1         X: attack 1.0973065
## 2    X: attplusheal 0.8451543
## 3 X: attack, health 0.8451543

This model seems to be even better since it allows the effect of health and attack to be different on the value of cost. We therefore chose to go with this model and try to adjust for additional effect imerged from card features. Since Hearthstone cards have descriptions on them and they are sometimes not quantifiable, we distinguished features that are easily quantifiable into categories. We ended up categorizing cards into Charge (cards can attack immediately once they were put on the stage), Divine Shield (cards have a protective shield that can protect them from reducing health during their first attack), Overload (specific cards for Shamman that can cause dramatic decrease in health at very early stage, but playing overload cards we limit the amount of mana players can use in the next round), Taunt (cards that can protect the hero, the opponent must attack taunts first before attacking the hero), Stealth (cards that are invisible and can not be attacked until their first attack), and Windfury (cards that can attack twice each turn). Cards that cannot be classified into these categories was then treated as normal cards and set to be reference group in the model.

## X: attack, health, mechanics(factors)
## Y: cost

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack + factor(mechanics1), cumulative(parallel = T, reverse = F), data = train)
summary(fitCL)
## 
## Call:
## vglm(formula = cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack + 
##     factor(mechanics1), family = cumulative(parallel = T, reverse = F), 
##     data = train)
## 
## Pearson residuals:
##                    Min        1Q    Median        3Q    Max
## logit(P[Y<=1])  -1.711 -0.150088 -0.046451 -0.009544  3.330
## logit(P[Y<=2])  -2.880 -0.217016 -0.047421  0.151582  9.053
## logit(P[Y<=3]) -12.558 -0.205327 -0.008254  0.243679  9.857
## logit(P[Y<=4])  -6.926 -0.098060  0.056029  0.179723 18.236
## logit(P[Y<=5])  -3.450  0.008930  0.041767  0.152892 21.704
## logit(P[Y<=6]) -28.417  0.008527  0.025442  0.084767 19.170
## 
## Coefficients:
##                                 Estimate Std. Error z value Pr(>|z|)    
## (Intercept):1                    2.80272    0.31293   8.956  < 2e-16 ***
## (Intercept):2                    5.06852    0.34258  14.795  < 2e-16 ***
## (Intercept):3                    7.01085    0.40926  17.130  < 2e-16 ***
## (Intercept):4                    8.93188    0.49063  18.205  < 2e-16 ***
## (Intercept):5                   10.70372    0.57072  18.755  < 2e-16 ***
## (Intercept):6                   12.97561    0.69643  18.631  < 2e-16 ***
## health                          -1.03594    0.07504 -13.804  < 2e-16 ***
## attack                          -1.05071    0.07585 -13.852  < 2e-16 ***
## factor(mechanics1)Charge        -2.07519    0.54697  -3.794 0.000148 ***
## factor(mechanics1)Divine Shield -1.18730    0.77100  -1.540 0.123574    
## factor(mechanics1)Overload       2.01039    1.11281   1.807 0.070827 .  
## factor(mechanics1)Stealth        0.19812    0.66840   0.296 0.766918    
## factor(mechanics1)Taunt          0.36004    0.38950   0.924 0.355292    
## factor(mechanics1)Windfury       0.41925    0.86627   0.484 0.628411    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Number of linear predictors:  6 
## 
## Dispersion Parameter for cumulative family:   1
## 
## Residual deviance: 1039.403 on 2620 degrees of freedom
## 
## Log-likelihood: -519.7015 on 2620 degrees of freedom
## 
## Number of iterations: 7 
## 
## Exponentiated coefficients:
##                          health                          attack 
##                       0.3548941                       0.3496877 
##        factor(mechanics1)Charge factor(mechanics1)Divine Shield 
##                       0.1255327                       0.3050445 
##      factor(mechanics1)Overload       factor(mechanics1)Stealth 
##                       7.4662076                       1.2191085 
##         factor(mechanics1)Taunt      factor(mechanics1)Windfury 
##                       1.4333910                       1.5208129

From the above output, we can see that after adjusting for health and attack, charge and overload are the two features that likely influenced the overall valuation model. We then considered a model which included 1) Charge, and 2) Charge and Overload.

## X: attack, health, charge
## Y: cost

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack + Charge, cumulative(parallel = T, reverse = F), data = train)
# summary(fitCL)


for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}


# To estimate the cost of cards based on attack, health, and charge:
test3 = test %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge)/(1+exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge)/(1+exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge)/(1+exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge)/(1+exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge)/(1+exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge)/(1+exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge))) - p1 - p2 - p3 - p4 - p5) %>% mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

test3 = test %>%
  select(cost, cost1, attack, health, cardId, playerClass, mechanics) %>% 
  left_join(test3, by = "cardId")
model3 = RMSE(test3$cost1, test3$value)
rmse_results = bind_rows(rmse_results,data_frame(method="X: attack, health, charge",  
                                     RMSE = model3))
rmse_results
## Source: local data frame [4 x 2]
## 
##                      method      RMSE
##                       (chr)     (dbl)
## 1                 X: attack 1.0973065
## 2            X: attplusheal 0.8451543
## 3         X: attack, health 0.8451543
## 4 X: attack, health, charge 0.8806306
## X: attack, health, charge, and overload
## Y: cost

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack + Charge + Overload, cumulative(parallel = T, reverse = F), data = train)
# summary(fitCL)


for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}


# To estimate the cost of cards based on attack, health, charge and overload:
test3 = test %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3 - p4 - p5) %>% mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

test3 = test %>%
  select(cost, cost1, attack, health, cardId, playerClass, mechanics) %>% 
  left_join(test3, by = "cardId")
model3 = RMSE(test3$cost1, test3$value)
rmse_results = bind_rows(rmse_results,data_frame(method="X: attack, health, charge, overload",  
                                     RMSE = model3))
rmse_results
## Source: local data frame [5 x 2]
## 
##                                method      RMSE
##                                 (chr)     (dbl)
## 1                           X: attack 1.0973065
## 2                      X: attplusheal 0.8451543
## 3                   X: attack, health 0.8451543
## 4           X: attack, health, charge 0.8806306
## 5 X: attack, health, charge, overload 0.8921426

We ended up using a model with the following covariates: health, attack, charge, overload:

final = minions_text1 %>% 
  select(cardId, name, cost,cost1,mechanics,Charge, Overload, attack, health) %>%
  rbind(highcost_card) %>%
  mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3 - p4 - p5) %>% 
  mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId, cost1, attack, health, name, mechanics) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
            value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
            value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
            value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
            value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
            value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

final %>% filter(value != cost1) %>%
  mutate(resid = value - cost1) %>%
  ggplot(aes(resid, group = mechanics, fill = mechanics)) + stat_bin(aes(y = ..count..), bins = 10 , position='dodge')

From the above plot, we can see that after adjusting for the claimed features (health, attack, charge, overload) of cards, some of the cards were overvalued (resid = estimated cost - assigned < 0) and some of them were undervalued (resid > 0). Cards that are Battlecry (cards that launch certain descriptive effects when cards are played), Normal (cards that do not have any special effects), and Deathrattle (cards that launch certain descriptive effects when cards are dead) are more frequently over- or under-valued.

Model validation: Blizard’s adjustment of card values:

Blizard company just annouced some adjustments of values of the following cards: Since we only focused on the valuation of minions cards (cards that can stay on the stage and have attack and health), this model can also be applied to minions:

new_assigned_cards = c("Knife Juggler", "Gig Game Hunter", "Force of Nature", 
  "Molten Giant", "Arcane Golem", "Blade Flurry", 
  "Keeper of the Grove", "Ancient of Lore", "Master of Disquise", 
  "Hunter's Mark", "Ironbeak Owl", "Leper Gnome")

final %>% filter(name %in% new_assigned_cards)
## Source: local data frame [7 x 7]
## Groups: cardId, cost1, attack, health, name [7]
## 
##     cardId cost1 attack health                name   mechanics value
##     (fctr) (dbl)  (int)  (int)              (fctr)       (chr) (dbl)
## 1  CS2_203     2      2      1        Ironbeak Owl   Battlecry     2
## 2  EX1_029     1      2      1         Leper Gnome Deathrattle     2
## 3  EX1_089     3      4      2        Arcane Golem      Charge     4
## 4  EX1_166     4      2      4 Keeper of the Grove      Normal     3
## 5  EX1_620     7      8      8        Molten Giant      Normal     7
## 6 NEW1_008     7      5      5     Ancient of Lore      Normal     5
## 7 NEW1_019     2      3      2       Knife Juggler      Normal     3

Ji, can you write some comments here?

Simulation

2. What is the balance between small cost cards and big cost cards? Assumptions: 1. Players will not use the card with cost 0 in the earlier several turns. 2. Cost can roughly represent the value of the card, thus we can maximum the cost of all 30 cards to maximum the value. 3. We focus on the first 5 turns.

First, create decks with all reasonable combinations of small cards (1-5) and others.

decks <- expand.grid(n1=0:6, n2=0:6, n3=0:6, n4=0:6, n5=0:6)
decks <- decks %>% tbl_df %>% mutate(others = 30-n1-n2-n3-n4-n5)

Next, use similation to estimate the probability to use card in the first 1/2/3/4/5-turn for each deck. Estimations are made for offensive player, as the defensive player has higher possiblity to use cards (4 cards at the begining with a special 0 cost card that temporatily increases the mana by 1) for the first few turns.

prob_usecard <- function(deck){
        card <- rep(c(1,2,3,4,5,10), deck)
                
        # offensive player
        temp <- t(replicate(1000,sample(card,30)))
        # assume choosing the 3 smallest cards for the starting hand
        sortcard <- t(apply(temp[,1:6],1,sort))
        temp[,1:6] <- sortcard
        sortcard2 <- t(apply(temp[,4:30],1,function(x){sample(x,27)}))
        temp[,4:30] <- sortcard2
        rm(sortcard)
        rm(sortcard2)
        
        # p1: can use card in the first turn
        p1 <- mean(apply(temp[,1:4],1,function(c){as.numeric(sum(c<2)>0)}))  
        # p2: can use card in the first 2 turns
        p2 <- mean(apply(temp[,1:5],1,function(c){as.numeric(sum(c<3)>0)}))  
        # p3: can use card in the first 3 turns
        p3 <- mean(apply(temp[,1:6],1,function(c){as.numeric(sum(c<4)>0)}))  
        # p4: can use card in the first 4 turns
        p4 <- mean(apply(temp[,1:7],1,function(c){as.numeric(sum(c<5)>0)}))  
        # p5: can use card in the first 5 turns
        p5 <- mean(apply(temp[,1:8],1,function(c){as.numeric(sum(c<6)>0)}))  
        
        c(p1, p2, p3, p4, p5)     
}

# get the probability of using card and combine
usecard <- t(apply(decks,1,prob_usecard))
colnames(usecard) <- c("p1","p2","p3","p4","p5")
decks <- cbind(decks,usecard) %>% 
        # add the total cost for each deck
        mutate(sum = n1+2*n2+3*n3+4*n4+5*n5+10*others) 
rm(usecard)

# save simulation results 
write.csv(decks,file="/Users/Yinnan/Desktop/2016/HearthScience/simulation.csv")
# get the simulation result from github
url <- "https://raw.githubusercontent.com/jihua0125/HearthScience/master/simulation.csv"
decks <- read_csv(url)

decks <- decks[,-1]

# constrain on probability of using card
decks.constrain <- decks %>% tbl_df %>% filter(p4>0.95, p2>0.5, p3>0.9, others>10) %>%
        arrange(desc(sum)) 

decks.constrain %>% summarize(min2 = min(n1+n2), min3 = min(n1+n2+n3), min4 = min(n1+n2+n3+n4))
## Source: local data frame [1 x 3]
## 
##    min2  min3  min4
##   (int) (int) (int)
## 1     2     6     7

Deck Analysis

3. Are there any “core” combination of cards? Instead of looking at the card information alone, we are trying to consider how one card interacts with others. We are using the built-up decks from top players of Hearthstone from the following website: http://www.hearthstonetopdecks.com/ A typical deck looks like this: [deck][deck.png]

Data Wrangling

classes<-c("druid/","hunter/","mage/","paladin/","priest/","rogue/","shaman/","warlock/","warrior/")
removeList<-c(9,6,10,10,4,7,10,7,7)
baseURL<-"http://www.hearthstonetopdecks.com/deck-category/class/"


totalInfoDeckList<-list()

heroDeckLists<-list()

for(k in 1:length(classes)){
  class<-classes[k]
  classBaseURL<-paste(baseURL,class,"page/",sep="")
  allDecks<-list()
  for (j in 1:5){
    tableURL<-paste(classBaseURL,j,sep="")
    tables<-as.data.frame(readHTMLTable(tableURL))
    deckNames<-lapply(tables[,2],as.character)
    deckNames<-unlist(deckNames)
    
    for(i in 1:length(deckNames)){
      urlName<-tolower(gsub("\\s","-",gsub("[^\\w \\s]+","",deckNames[i],perl = TRUE),perl = TRUE))
      
      testURL<-paste("http://www.hearthstonetopdecks.com/decks/",urlName,sep="")
      tryCatch(webpage<-read_html(testURL),error=function(e){return(i)})
      
      cardNames<-webpage%>%
        html_nodes(".card-name")%>%
        html_text()
      
      cardCounts<-webpage%>%
        html_nodes(".card-count")%>%
        html_text()%>%
        as.numeric()
      
      deckId<-(j-1)*25+i
      
      deck<-cbind(cardNames,cardCounts,rep(deckId,length(cardNames)))
      
      allDecks[[deckId]]<-deck
    }
  }
  largerTable<-data.frame()
  
  for (i in removeList[k]:125){
    largerTable<-rbind(largerTable,allDecks[[i]])
  }
  
  largerTable<-largerTable%>%spread(key=V3,value=cardCounts)
  
  for (i in 2:length(largerTable)){
    largerTable[,i]<-as.numeric(as.character(largerTable[,i]))
  }
  
  largerTable[is.na(largerTable)]<-0
  
  heroDeckLists[[k]]<-largerTable
  
}

for(i in 1:9){
  totalInfoDeckList[[i]]<-heroDeckLists[[i]]%>%select(c(1,length(heroDeckLists[[i]])))
}

for(i in 1:9){
  totalInfoDeckList[[i]]<-totalInfoDeckList[[i]]%>%left_join(cards,by=c("cardNames"="name"))
}
decks<-list()
for(i in 1:9){
  decks[[i]]<-heroDeckLists[[i]]%>%gather(deckId,cardCounts,2:(length(heroDeckLists[[i]])-1))
}

Deeper look into the deck information

From our empirical knowledge, we know that each deck has its own strategy to win the game, such as aggro, control, midrange, face, etc. These strategies are highly related to the average cost of all the minions inside the deck.

minions<-read.csv("minions.csv",sep="\t")
weapons<-read.csv("weapons.csv",sep="\t")
spells<-read.csv("spells.csv",sep="\t")
cards<-rbind(minions,weapons,spells)
# load("D:/HSPH/BIO 260/final/data/minions_text.RData")

classes<-c("druid","hunter","mage","paladin","priest","rogue","shaman","warlock","warrior")
decks<-list()
heroDeckLists<-list()
for(i in 1:9){
  filename<-paste(classes[i],"decks.csv",sep="")
  heroDeckLists[[i]]<-read.csv(filename,sep="\t")
  decks[[i]]<-heroDeckLists[[i]]%>%gather(deckId,cardCounts,2:(length(heroDeckLists[[i]])-1))
}

###warlock deck
warlockDeckCost<-decks[[8]]%>%filter(cardCounts!=0)%>%
  left_join(cards,by=c("cardNames"="name"))%>%
  filter(type=="Minion")%>%
  group_by(deckId)%>%
  mutate(cardTotalCost=cost*cardCounts)%>%
  mutate(aveCost=mean(cardTotalCost))%>%
  ungroup()

warlockDeckCost%>%select(deckId,aveCost)%>%distinct()%>%
  ggplot(aes(x=aveCost))+geom_histogram(binwidth = 0.5)+ggtitle("Warlock deck distribution")

###paladin deck
paladinDeckCost<-decks[[4]]%>%filter(cardCounts!=0)%>%
  left_join(cards,by=c("cardNames"="name"))%>%
  filter(type=="Minion")%>%
  group_by(deckId)%>%
  mutate(cardTotalCost=cost*cardCounts)%>%
  mutate(aveCost=mean(cardTotalCost))%>%
  ungroup()

paladinDeckCost%>%select(deckId,aveCost)%>%distinct()%>%
  ggplot(aes(x=aveCost))+geom_histogram(binwidth = 0.5)+ggtitle("Paladin deck distribution")

###druid deck
druidDeckCost<-decks[[1]]%>%filter(cardCounts!=0)%>%
  left_join(cards,by=c("cardNames"="name"))%>%
  filter(type=="Minion")%>%
  group_by(deckId)%>%
  mutate(cardTotalCost=cost*cardCounts)%>%
  mutate(aveCost=mean(cardTotalCost))%>%
  ungroup()

druidDeckCost%>%select(deckId,aveCost)%>%distinct()%>%
  ggplot(aes(x=aveCost))+geom_histogram(binwidth = 0.5)+ggtitle("Druid deck distribution")

###hunter deck
hunterDeckCost<-decks[[2]]%>%filter(cardCounts!=0)%>%
  left_join(cards,by=c("cardNames"="name"))%>%
  filter(type=="Minion")%>%
  group_by(deckId)%>%
  mutate(cardTotalCost=cost*cardCounts)%>%
  mutate(aveCost=mean(cardTotalCost))%>%
  ungroup()

hunterDeckCost%>%select(deckId,aveCost)%>%distinct()%>%
  ggplot(aes(x=aveCost))+geom_histogram(binwidth = 0.5)+ggtitle("Hunter deck distribution")

###Mage deck
mageDeckCost<-decks[[3]]%>%filter(cardCounts!=0)%>%
  left_join(cards,by=c("cardNames"="name"))%>%
  filter(type=="Minion")%>%
  group_by(deckId)%>%
  mutate(cardTotalCost=cost*cardCounts)%>%
  mutate(aveCost=mean(cardTotalCost))%>%
  ungroup()

mageDeckCost%>%select(deckId,aveCost)%>%distinct()%>%
  ggplot(aes(x=aveCost))+geom_histogram(binwidth = 0.5)+ggtitle("Mage deck distribution")

###Priest deck
priestDeckCost<-decks[[5]]%>%filter(cardCounts!=0)%>%
  left_join(cards,by=c("cardNames"="name"))%>%
  filter(type=="Minion")%>%
  group_by(deckId)%>%
  mutate(cardTotalCost=cost*cardCounts)%>%
  mutate(aveCost=mean(cardTotalCost))%>%
  ungroup()

priestDeckCost%>%select(deckId,aveCost)%>%distinct()%>%
  ggplot(aes(x=aveCost))+geom_histogram(binwidth = 0.5)+ggtitle("Priest deck distribution")

##Rogue deck
rogueDeckCost<-decks[[6]]%>%filter(cardCounts!=0)%>%
  left_join(cards,by=c("cardNames"="name"))%>%
  filter(type=="Minion")%>%
  group_by(deckId)%>%
  mutate(cardTotalCost=cost*cardCounts)%>%
  mutate(aveCost=mean(cardTotalCost))%>%
  ungroup()

rogueDeckCost%>%select(deckId,aveCost)%>%distinct()%>%
  ggplot(aes(x=aveCost))+geom_histogram(binwidth = 0.5)+ggtitle("Rogue deck distribution")

###Shaman
shamanDeckCost<-decks[[7]]%>%filter(cardCounts!=0)%>%
  left_join(cards,by=c("cardNames"="name"))%>%
  filter(type=="Minion")%>%
  group_by(deckId)%>%
  mutate(cardTotalCost=cost*cardCounts)%>%
  mutate(aveCost=mean(cardTotalCost))%>%
  ungroup()

shamanDeckCost%>%select(deckId,aveCost)%>%distinct()%>%
  ggplot(aes(x=aveCost))+geom_histogram(binwidth = 0.5)+ggtitle("Shaman deck distribution")

###Warrior deck
warriorDeckCost<-decks[[9]]%>%filter(cardCounts!=0)%>%
  left_join(cards,by=c("cardNames"="name"))%>%
  filter(type=="Minion")%>%
  group_by(deckId)%>%
  mutate(cardTotalCost=cost*cardCounts)%>%
  mutate(aveCost=mean(cardTotalCost))%>%
  ungroup()

warriorDeckCost%>%select(deckId,aveCost)%>%distinct()%>%
  ggplot(aes(x=aveCost))+geom_histogram(binwidth = 0.5)+ggtitle("Warrior deck distribution")

###summary
druidDeckCost%>%select(deckId,aveCost)%>%distinct()%>%summary()%>%kable
deckId aveCost
Length:109 Min. : 4.000
Class :character 1st Qu.: 6.769
Mode :character Median : 7.067
NA Mean : 7.209
NA 3rd Qu.: 7.700
NA Max. :12.143 
hunterDeckCost%>%select(deckId,aveCost)%>%distinct()%>%summary()%>%kable
deckId aveCost
Length:119 Min. :3.111
Class :character 1st Qu.:3.600
Mode :character Median :4.900
NA Mean :4.766
NA 3rd Qu.:5.600
NA Max. :7.250 
mageDeckCost%>%select(deckId,aveCost)%>%distinct()%>%summary()%>%kable
deckId aveCost
Length:115 Min. :3.500
Class :character 1st Qu.:4.917
Mode :character Median :5.300
NA Mean :5.582
NA 3rd Qu.:6.091
NA Max. :8.833 
paladinDeckCost%>%select(deckId,aveCost)%>%distinct()%>%summary()%>%kable
deckId aveCost
Length:115 Min. : 2.900
Class :character 1st Qu.: 4.930
Mode :character Median : 5.533
NA Mean : 5.387
NA 3rd Qu.: 5.905
NA Max. :11.000 
priestDeckCost%>%select(deckId,aveCost)%>%distinct()%>%summary()%>%kable
deckId aveCost
Length:121 Min. : 4.444
Class :character 1st Qu.: 5.429
Mode :character Median : 5.786
NA Mean : 6.012
NA 3rd Qu.: 6.364
NA Max. :12.833 
rogueDeckCost%>%select(deckId,aveCost)%>%distinct()%>%summary()%>%kable
deckId aveCost
Length:118 Min. :3.167
Class :character 1st Qu.:5.111
Mode :character Median :5.333
NA Mean :5.578
NA 3rd Qu.:6.000
NA Max. :9.778 
shamanDeckCost%>%select(deckId,aveCost)%>%distinct()%>%summary()%>%kable
deckId aveCost
Length:115 Min. : 2.556
Class :character 1st Qu.: 5.500
Mode :character Median : 6.000
NA Mean : 5.864
NA 3rd Qu.: 6.481
NA Max. :10.500 
warlockDeckCost%>%select(deckId,aveCost)%>%distinct()%>%summary()%>%kable
deckId aveCost
Length:118 Min. : 3.467
Class :character 1st Qu.: 4.420
Mode :character Median : 5.426
NA Mean : 6.277
NA 3rd Qu.: 8.765
NA Max. :10.615 
warriorDeckCost%>%select(deckId,aveCost)%>%distinct()%>%summary()%>%kable
deckId aveCost
Length:118 Min. :4.368
Class :character 1st Qu.:5.700
Mode :character Median :6.000
NA Mean :6.086
NA 3rd Qu.:6.692
NA Max. :8.000
From the histogram we ca n see warlock is quite different from other heros, the distribution of the costs of decks has double peaks, while others are more likely following a normal distribution. This finding gives us a suggestion to explore data furtherly.

Explore Card Interactions within Warlock Decks

Let’s have a look at the correlation between the cards within warlock decks.

data<-read.csv("correlation.csv")
colnames(data)<-gsub("\\."," ",colnames(data))
#warlockDecks<-heroDeckLists[[8]]
#rownames(warlockDecks)<-t(warlockDecks[,1])
#data<-warlockDecks%>%select(-cardNames)

#calculate correlation matrix
corMatrix<-cor(x=data)
hClust<-hclust(dist(data),method="complete")
plot(hClust,cex=0.6)

pc<-prcomp(corMatrix)
summary(pc)
## Importance of components:
##                           PC1     PC2     PC3    PC4     PC5    PC6
## Standard deviation     1.9738 0.81496 0.69388 0.6568 0.53860 0.4793
## Proportion of Variance 0.5581 0.09514 0.06897 0.0618 0.04156 0.0329
## Cumulative Proportion  0.5581 0.65324 0.72221 0.7840 0.82557 0.8585
##                            PC7     PC8     PC9    PC10    PC11    PC12
## Standard deviation     0.39575 0.35725 0.33647 0.29937 0.28180 0.24043
## Proportion of Variance 0.02244 0.01828 0.01622 0.01284 0.01138 0.00828
## Cumulative Proportion  0.88091 0.89919 0.91541 0.92824 0.93962 0.94790
##                           PC13    PC14    PC15    PC16    PC17    PC18
## Standard deviation     0.23998 0.18815 0.18349 0.16857 0.16221 0.14772
## Proportion of Variance 0.00825 0.00507 0.00482 0.00407 0.00377 0.00313
## Cumulative Proportion  0.95615 0.96122 0.96605 0.97012 0.97389 0.97701
##                           PC19    PC20    PC21    PC22    PC23    PC24
## Standard deviation     0.14260 0.12982 0.12585 0.11783 0.11170 0.10653
## Proportion of Variance 0.00291 0.00241 0.00227 0.00199 0.00179 0.00163
## Cumulative Proportion  0.97993 0.98234 0.98461 0.98660 0.98838 0.99001
##                           PC25    PC26    PC27    PC28    PC29    PC30
## Standard deviation     0.10423 0.09051 0.08557 0.08310 0.07612 0.06852
## Proportion of Variance 0.00156 0.00117 0.00105 0.00099 0.00083 0.00067
## Cumulative Proportion  0.99157 0.99274 0.99379 0.99478 0.99561 0.99628
##                           PC31    PC32    PC33    PC34    PC35    PC36
## Standard deviation     0.06737 0.06297 0.05612 0.05256 0.04522 0.03921
## Proportion of Variance 0.00065 0.00057 0.00045 0.00040 0.00029 0.00022
## Cumulative Proportion  0.99693 0.99750 0.99795 0.99835 0.99864 0.99886
##                           PC37    PC38    PC39    PC40    PC41    PC42
## Standard deviation     0.03380 0.03334 0.03195 0.03020 0.02749 0.02258
## Proportion of Variance 0.00016 0.00016 0.00015 0.00013 0.00011 0.00007
## Cumulative Proportion  0.99902 0.99918 0.99933 0.99946 0.99957 0.99964
##                           PC43    PC44    PC45    PC46    PC47    PC48
## Standard deviation     0.02142 0.01927 0.01714 0.01688 0.01367 0.01317
## Proportion of Variance 0.00007 0.00005 0.00004 0.00004 0.00003 0.00002
## Cumulative Proportion  0.99971 0.99976 0.99980 0.99984 0.99987 0.99989
##                           PC49    PC50    PC51     PC52     PC53     PC54
## Standard deviation     0.01275 0.01135 0.01054 0.009396 0.008152 0.007593
## Proportion of Variance 0.00002 0.00002 0.00002 0.000010 0.000010 0.000010
## Cumulative Proportion  0.99992 0.99994 0.99995 0.999960 0.999970 0.999980
##                            PC55    PC56     PC57     PC58     PC59
## Standard deviation     0.005803 0.00479 0.004449 0.003708 0.003085
## Proportion of Variance 0.000000 0.00000 0.000000 0.000000 0.000000
## Cumulative Proportion  0.999990 0.99999 0.999990 0.999990 1.000000
##                            PC60     PC61     PC62     PC63     PC64
## Standard deviation     0.002767 0.002548 0.002223 0.001594 0.001348
## Proportion of Variance 0.000000 0.000000 0.000000 0.000000 0.000000
## Cumulative Proportion  1.000000 1.000000 1.000000 1.000000 1.000000
##                            PC65      PC66      PC67      PC68      PC69
## Standard deviation     0.001251 0.0009424 0.0009018 0.0006618 0.0005387
## Proportion of Variance 0.000000 0.0000000 0.0000000 0.0000000 0.0000000
## Cumulative Proportion  1.000000 1.0000000 1.0000000 1.0000000 1.0000000
##                             PC70     PC71      PC72      PC73      PC74
## Standard deviation     0.0004562 0.000395 0.0002827 0.0002188 0.0001501
## Proportion of Variance 0.0000000 0.000000 0.0000000 0.0000000 0.0000000
## Cumulative Proportion  1.0000000 1.000000 1.0000000 1.0000000 1.0000000
##                             PC75      PC76      PC77      PC78      PC79
## Standard deviation     9.085e-05 4.207e-05 6.774e-16 1.884e-16 1.884e-16
## Proportion of Variance 0.000e+00 0.000e+00 0.000e+00 0.000e+00 0.000e+00
## Cumulative Proportion  1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00
##                             PC80      PC81      PC82      PC83      PC84
## Standard deviation     1.884e-16 1.884e-16 1.884e-16 1.884e-16 1.884e-16
## Proportion of Variance 0.000e+00 0.000e+00 0.000e+00 0.000e+00 0.000e+00
## Cumulative Proportion  1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00
##                             PC85      PC86      PC87      PC88      PC89
## Standard deviation     1.884e-16 1.884e-16 1.884e-16 1.884e-16 1.884e-16
## Proportion of Variance 0.000e+00 0.000e+00 0.000e+00 0.000e+00 0.000e+00
## Cumulative Proportion  1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00
##                             PC90      PC91      PC92      PC93      PC94
## Standard deviation     1.884e-16 1.884e-16 1.884e-16 1.884e-16 1.884e-16
## Proportion of Variance 0.000e+00 0.000e+00 0.000e+00 0.000e+00 0.000e+00
## Cumulative Proportion  1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00
##                             PC95      PC96      PC97      PC98      PC99
## Standard deviation     1.884e-16 1.884e-16 1.884e-16 1.884e-16 1.884e-16
## Proportion of Variance 0.000e+00 0.000e+00 0.000e+00 0.000e+00 0.000e+00
## Cumulative Proportion  1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00
##                            PC100     PC101     PC102
## Standard deviation     1.884e-16 1.884e-16 8.069e-17
## Proportion of Variance 0.000e+00 0.000e+00 0.000e+00
## Cumulative Proportion  1.000e+00 1.000e+00 1.000e+00
data.t<-t(data)
d1<-dist(data) 
d2<-dist(data.t)
cormat<-round(cor(data.t),2)
mtscaled<-as.matrix(d1)


### triangle heatmap

source("https://raw.githubusercontent.com/briatte/ggcorr/master/ggcorr.R")
##ggcorr(cormat)
ggcorr(cormat,hjust = 0.3, size = 1, color = "grey50")

From the principle components analysis, we can see the top 2 principle components have explained 2/3 of the variance between cards. So here, we are going to use the first 2 pcs to do the following analysisto keep the scale of problem small enough.

pcaData <-pc$x[,1:9]
pca1 <-pc$x[,1]
pca2 <-pc$x[,2]
pca3<- pc$x[,3]
pca4 <-pc$x[,4]
pca5 <-pc$x[,5]
pca6<- pc$x[,6]
pca7 <-pc$x[,7]
pca8 <-pc$x[,8]
pca9<- pc$x[,9]

wss <- (nrow(pcaData)-1)*sum(apply(pcaData,2,var))
for (i in 2:20) wss[i] <- sum(kmeans(pcaData,centers=i)$withinss)
plot(1:20, wss, type="b", xlab="Number of Clusters",
     ylab="Within groups sum of squares")

kmeans.cluster<-kmeans(pcaData, centers=4)
pc.df<-data.frame(ID=names(pca1),PCA1=pca1, PCA2=pca2, PCA3=pca3,PCA4=pca4,PCA5=pca5,PCA6=pca6,PCA7=pca7,PCA8=pca8,PCA9=pca9, Cluster=factor(kmeans.cluster$cluster))
pc.df%>%ggplot(aes(x=PCA1, y=PCA2, label=ID, color=Cluster))+geom_jitter()+
  geom_text_repel(aes(PCA1, PCA2, label=ID),data = filter(pc.df,PCA1 < -2.5 | PCA1 >2.5| PCA2 < -1.5 | PCA2>1.5))

total.df<-pc.df%>%left_join(cards,by=c("ID"="name"))
total.df%>%ggplot(aes(x=PCA1, y=PCA2, label=cost, color=Cluster))+geom_jitter()+geom_text_repel()

pc.df%>%group_by(Cluster)%>%summarize(n())
## Source: local data frame [4 x 2]
## 
##   Cluster   n()
##    (fctr) (int)
## 1       1    19
## 2       2    56
## 3       3    17
## 4       4    10

look at 4 clusters

In the above codes, we have tried to use Kmeans clustering to distinguish different type of decks. By the FOM plots, we found that 4 is the balanced point, so we made a 4 centroid clustering. Let’s pick one deck to see if this clustering make sense.

deck<-heroDeckLists[[8]]%>%select(cardNames,X60)%>%
  filter(X60!=0)%>%
  left_join(pc.df,by=c("cardNames"="ID"))
deck[,c(1,12)]%>%kable
cardNames Cluster
Abusive Sergeant 3
Dark Peddler 3
Defender of Argus 3
Flame Imp 3
Imp Gang Boss 3
Knife Juggler 3
Voidwalker 3
Hellfire 1
Loatheb 2
Haunted Creeper 3
Nerubian Egg 3
Power Overwhelming 3
Doomguard 3
Soulfire 4
Fist of Jaraxxus 2
Leper Gnome 2 
### seperate data set
fullcluster<-pc.df%>%select(-PCA1,-PCA2)
cluster1<-fullcluster%>%filter(Cluster=="1")%>%select(-Cluster)
cluster2<-fullcluster%>%filter(Cluster=="2")%>%select(-Cluster)
cluster3<-fullcluster%>%filter(Cluster=="3")%>%select(-Cluster)
cluster4<-fullcluster%>%filter(Cluster=="4")%>%select(-Cluster)
#conver the rownames to first column "ID"
ID<-rownames(data)
rownames(data)<-NULL
data<-cbind(ID,data)
#create 4 dataset by "ID"
dataset1<-dplyr::right_join(data,cluster1,by="ID")
dataset2<-dplyr::right_join(data,cluster2,by="ID")
dataset3<-dplyr::right_join(data,cluster3,by="ID")
dataset4<-dplyr::right_join(data,cluster4,by="ID")

#convert the first column to rownames
rownames(dataset1)<-dataset1$ID
rownames(dataset2)<-dataset2$ID
rownames(dataset3)<-dataset3$ID
rownames(dataset4)<-dataset4$ID
dataset1<-dataset1[,-1]
dataset2<-dataset2[,-1]
dataset3<-dataset3[,-1]
dataset4<-dataset4[,-1]
data1.t<-t(dataset1)
data2.t<-t(dataset2)
data3.t<-t(dataset3)
data4.t<-t(dataset4)
#correlation within the first dataset
cormat1<-round(cor(data1.t),2)
cormat2<-round(cor(data2.t),2)
cormat3<-round(cor(data3.t),2)
cormat4<-round(cor(data4.t),2)
# HC of the first dataset
#hClust1<-hclust(dist(dataset1),method="complete")
#hClust2<-hclust(dist(dataset2),method="complete")
#hClust3<-hclust(dist(dataset3),method="complete")
#hClust4<-hclust(dist(dataset4),method="complete")
#plot(hClust1,cex=0.6)
#plot(hClust2,cex=0.6)
#plot(hClust3,cex=0.6)
#plot(hClust4,cex=0.6)

#correlation matrix 
melted_cormat1 <- melt(cormat1)
p1<-ggplot(data = melted_cormat1, aes(X2, X1, fill = value))+
  geom_tile(color = "white")+
  scale_fill_gradient2(low = "blue", high = "red", mid = "white", 
                       midpoint = 0, limit = c(-1,1), space = "Lab", 
                       name="Pearson\nCorrelation") +
  theme_minimal()

p1+ theme(axis.text.y = element_text(vjust = 1, 
                                     size = 4, hjust = 1))+theme(axis.text.x = element_text(angle = 45, vjust = 1, 
                                                                                             size = 3, hjust = 1))+scale_x_discrete(expand = c(0, 0)) + scale_y_discrete(expand = c(0, 0))+theme(legend.background=element_rect(),legend.margin=unit(1,"cm"))

melted_cormat2 <- melt(cormat2)
p2<-ggplot(data = melted_cormat2, aes(X2, X1, fill = value))+
  geom_tile(color = "white")+
  scale_fill_gradient2(low = "blue", high = "red", mid = "white", 
                       midpoint = 0, limit = c(-1,1), space = "Lab", 
                       name="Pearson\nCorrelation") +
  theme_minimal()

p2+ theme(axis.text.y = element_text(vjust = 1, 
                                     size = 4, hjust = 1))+theme(axis.text.x = element_text(angle = 45, vjust = 1, 
                                                                                             size = 10, hjust = 1))+scale_x_discrete(expand = c(0, 0)) + scale_y_discrete(expand = c(0, 0))+theme(legend.background=element_rect(),legend.margin=unit(1,"cm"))


melted_cormat3 <- melt(cormat3)
p3<-ggplot(data = melted_cormat3, aes(X2, X1, fill = value))+
  geom_tile(color = "white")+
  scale_fill_gradient2(low = "blue", high = "red", mid = "white", 
                       midpoint = 0, limit = c(-1,1), space = "Lab", 
                       name="Pearson\nCorrelation") +
  theme_minimal()

p3+ theme(axis.text.y = element_text(vjust = 1, 
                                     size = 10, hjust = 1))+theme(axis.text.x = element_text(angle = 45, vjust = 1, 
                                                                                             size = 10, hjust = 1))+scale_x_discrete(expand = c(0, 0)) + scale_y_discrete(expand = c(0, 0))+theme(legend.background=element_rect(),legend.margin=unit(1,"cm"))


melted_cormat4 <- melt(cormat4)
p4<-ggplot(data = melted_cormat4, aes(X2, X1, fill = value))+
  geom_tile(color = "white")+
  scale_fill_gradient2(low = "blue", high = "red", mid = "white", 
                       midpoint = 0, limit = c(-1,1), space = "Lab", 
                       name="Pearson\nCorrelation") +
  theme_minimal()

p4+ theme(axis.text.y = element_text(vjust = 1, 
                                     size = 10, hjust = 1))+theme(axis.text.x = element_text(angle = 45, vjust = 1, 
                                                                                             size = 10, hjust = 1))+scale_x_discrete(expand = c(0, 0)) + scale_y_discrete(expand = c(0, 0))+theme(legend.background=element_rect(),legend.margin=unit(1,"cm"))

Let’s have a look at the card frequency distribution.

freqTable<-heroDeckLists[[8]]%>%tbl_df()%>%
  mutate(cardTotalCounts=rowSums(heroDeckLists[[8]][,2:length(heroDeckLists[[8]])]))%>%
  dplyr::select(cardNames,cardTotalCounts)
total.df<-total.df%>%left_join(freqTable,by=c("ID"="cardNames"))
total.df%>%dplyr::select(ID,cardTotalCounts,Cluster)%>%filter(complete.cases(.))%>%
  ggplot(aes(Cluster,cardTotalCounts))+geom_point()

From the above plots, we can see that the cards in cluster 1 and 4 are more frequent appear in decks.This helps us to select the core cards of a deck. A core card should neither appear too much, which makes it look like panacea; nor appear too little, which means it has fewer interaction with other cards.

coreTable<-total.df%>%filter(type=="Minion")%>%dplyr::select(ID,cardTotalCounts,Cluster,cost)%>%filter(complete.cases(.))%>%
  filter(cardTotalCounts<90&cardTotalCounts>60)

coreTable%>%group_by(Cluster)%>%summarize(n())
## Source: local data frame [2 x 2]
## 
##   Cluster   n()
##    (fctr) (int)
## 1       1     6
## 2       3     7

Now, in each cluster, we have several numbers of core cards. But 6 and 7 core cards are a bit too many. So let’s do a simulation of how numbers of core cards affect the probability of getting all the core cards after drawing certain amount of cards. #### Number of Draw cards For each deck, there are several “core” cards that can have the greatest effect when they are used together. We will usually put 2 cards for each component of core cards, and we want to get at least one for every component as early as possible.

First we list all possible decks with core cards and normal cards. Each set of core cards includes 2-5 different components. We consider the offensive side/early hand first.

# sort the first 6 card for offensive side/early hand, assume we will always keep the core card
sort.offensive <- function(tmp){
        sortcard <- t(apply(tmp[,1:6],1,function(x){sort(x,decreasing = T)}))
        tmp[,1:6] <- sortcard
        sortcard2 <- t(apply(tmp[,4:30],1,function(x){sample(x,27)}))
        tmp[,4:30] <- sortcard2
        tmp
}

# 2 components core cards set, each with 2 cards
card <- c(1,1,2,2,rep(0,26))
tmp <- t(replicate(10000,sample(card,30)))

tmp <- sort.offensive(tmp)

# the probability of having the complete set of core cards by n turns
# we want to know the probability by each turn i

offen_core2 <- function(i){
        mean(apply(tmp[,1:(i+3)],1,function(c){as.numeric(sum(c==1)>0 & sum(c==2)>0)})) 
}

o2 <- sapply(1:27,offen_core2)

# 3 components core cards set, each with 2 cards
card <- c(1,1,2,2,3,3,rep(0,24))
tmp <- t(replicate(10000,sample(card,30)))

tmp <- sort.offensive(tmp)

# the probability of having the complete set of core cards by n turns
# we want to know the probability by each turn i
offen_core3 <- function(i){
        mean(apply(tmp[,1:(i+3)],1,function(c){as.numeric(sum(c==1)>0 & sum(c==2)>0 & sum(c==3)>0)})) 
}

o3 <- sapply(1:27,offen_core3)


# 4 components core cards set, each with 2 cards
card <- c(1,1,2,2,3,3,4,4,rep(0,22))
tmp <- t(replicate(10000,sample(card,30)))

tmp <- sort.offensive(tmp)

# the probability of having the complete set of core cards by n turns
# we want to know the probability by each turn i
offen_core4 <- function(i){
        mean(apply(tmp[,1:(i+3)],1,function(c){as.numeric(sum(c==1)>0 & sum(c==2)>0 & sum(c==3)>0) & sum(c==4)>0})) 
}

o4 <- sapply(1:27,offen_core4) 


# 5 components core card set
card <- c(1,1,2,2,3,3,4,4,5,5,rep(0,20))
tmp <- t(replicate(10000,sample(card,30)))

tmp <- sort.offensive(tmp)

# the probability of having the complete set of core cards by n turns
# we want to know the probability by each turn i
offen_core5 <- function(i){
        mean(apply(tmp[,1:(i+3)],1,function(c){as.numeric(sum(c==1)>0 & sum(c==2)>0 & sum(c==3)>0) & sum(c==4)>0 & sum(c==5)>0})) 
}

o5 <- sapply(1:27,offen_core5) 

# 6 components core card set
card <- c(1,1,2,2,3,3,4,4,5,5,6,6,rep(0,18))
tmp <- t(replicate(10000,sample(card,30)))

tmp <- sort.offensive(tmp)

# the probability of having the complete set of core cards by n turns
# we want to know the probability by each turn i
offen_core6 <- function(i){
        mean(apply(tmp[,1:(i+3)],1,function(c){as.numeric(sum(c==1)>0 & sum(c==2)>0 & sum(c==3)>0) & sum(c==4)>0 & sum(c==5)>0 & sum(c==6)>0})) 
}

o6 <- sapply(1:27,offen_core6) 

# 7 components core card set
card <- c(1,1,2,2,3,3,4,4,5,5,6,6,7,7,rep(0,18))
tmp <- t(replicate(10000,sample(card,30)))

tmp <- sort.offensive(tmp)

# the probability of having the complete set of core cards by n turns
# we want to know the probability by each turn i
offen_core7 <- function(i){
        mean(apply(tmp[,1:(i+3)],1,function(c){as.numeric(sum(c==1)>0 & sum(c==2)>0 & sum(c==3)>0) & sum(c==4)>0 & sum(c==5)>0 & sum(c==6)>0 & sum(c==7)>0})) 
}

o7 <- sapply(1:27,offen_core7)

Similarly we can estimate the probability for the defensive side/late hand.

# sort the first 6 card for offensive side, assume we will always keep the core card
sort.defensive <- function(tmp){
        sortcard <- t(apply(tmp[,1:8],1,function(x){sort(x,decreasing = T)}))
        tmp[,1:8] <- sortcard
        sortcard2 <- t(apply(tmp[,5:30],1,function(x){sample(x,26)}))
        tmp[,5:30] <- sortcard2
        tmp
}

# 2 components core cards set, each with 2 cards
card <- c(1,1,2,2,rep(0,26))
tmp <- t(replicate(10000,sample(card,30)))

tmp <- sort.defensive(tmp)

# the probability of having the complete set of core cards by n turns
# we want to know the probability by each turn i

defen_core2 <- function(i){
        mean(apply(tmp[,1:(i+4)],1,function(c){as.numeric(sum(c==1)>0 & sum(c==2)>0)})) 
}

d2 <- sapply(1:26,defen_core2)

# 3 components core cards set, each with 2 cards
card <- c(1,1,2,2,3,3,rep(0,24))
tmp <- t(replicate(10000,sample(card,30)))

tmp <- sort.defensive(tmp)

# the probability of having the complete set of core cards by n turns
# we want to know the probability by each turn i
defen_core3 <- function(i){
        mean(apply(tmp[,1:(i+4)],1,function(c){as.numeric(sum(c==1)>0 & sum(c==2)>0 & sum(c==3)>0)})) 
}

d3 <- sapply(1:26,defen_core3)

# 4 components core cards set, each with 2 cards
card <- c(1,1,2,2,3,3,4,4,rep(0,22))
tmp <- t(replicate(10000,sample(card,30)))

tmp <- sort.defensive(tmp)

# the probability of having the complete set of core cards by n turns
# we want to know the probability by each turn i
defen_core4 <- function(i){
        mean(apply(tmp[,1:(i+3)],1,function(c){as.numeric(sum(c==1)>0 & sum(c==2)>0 & sum(c==3)>0) & sum(c==4)>0})) 
}

d4 <- sapply(1:26,defen_core4) 


# 5 components core card set
card <- c(1,1,2,2,3,3,4,4,5,5,rep(0,20))
tmp <- t(replicate(10000,sample(card,30)))

tmp <- sort.defensive(tmp)

# the probability of having the complete set of core cards by n turns
# we want to know the probability by each turn i
defen_core5 <- function(i){
        mean(apply(tmp[,1:(i+3)],1,function(c){as.numeric(sum(c==1)>0 & sum(c==2)>0 & sum(c==3)>0) & sum(c==4)>0 & sum(c==5)>0})) 
}

d5 <- sapply(1:26,defen_core5) 

# 6 components core card set
card <- c(1,1,2,2,3,3,4,4,5,5,6,6,rep(0,18))
tmp <- t(replicate(10000,sample(card,30)))

tmp <- sort.defensive(tmp)

# the probability of having the complete set of core cards by n turns
# we want to know the probability by each turn i
defen_core6 <- function(i){
        mean(apply(tmp[,1:(i+3)],1,function(c){as.numeric(sum(c==1)>0 & sum(c==2)>0 & sum(c==3)>0) & sum(c==4)>0 & sum(c==5)>0 & sum(c==6)>0})) 
}

d6 <- sapply(1:26,defen_core6)

# 7 components core card set
card <- c(1,1,2,2,3,3,4,4,5,5,6,6,7,7,rep(0,16))
tmp <- t(replicate(10000,sample(card,30)))

tmp <- sort.defensive(tmp)

# the probability of having the complete set of core cards by n turns
# we want to know the probability by each turn i
defen_core7 <- function(i){
        mean(apply(tmp[,1:(i+3)],1,function(c){as.numeric(sum(c==1)>0 & sum(c==2)>0 & sum(c==3)>0) & sum(c==4)>0 & sum(c==5)>0 & sum(c==6)>0 & sum(c==7)>0})) 
}

d7 <- sapply(1:26,defen_core7) 
# show results: the probability of getting the whole set of core cards

offensive <- data.frame(o2,o3,o4,o5,o6,o7)
colnames(offensive) <- c(2,3,4,5,6,7)
offensive <- offensive %>% mutate(turn=1:27, card=4:30) %>% gather("n_core","prob",1:6)

defensive <- data.frame(d2,d3,d4,d5,d6,d7)
colnames(defensive) <- c(2,3,4,5,6,7)
defensive <- defensive %>% mutate(turn=1:26, card=5:30) %>% gather("n_core","prob",1:6)

offensive %>% ggplot(aes(card,prob)) + geom_line(aes(color=n_core)) +
        ggtitle("Early hand") +
        scale_x_continuous(breaks=4:30) +
        scale_y_continuous(breaks=seq(0,1,0.1)) +
        geom_vline(xintercept = 13)

defensive %>% ggplot(aes(card,prob)) + geom_line(aes(color=n_core)) +
        ggtitle("Late hand") +
        scale_x_continuous(breaks=5:30) +
        scale_y_continuous(breaks=seq(0,1,0.1)) +
        geom_vline(xintercept = 14)

coreTable<-coreTable%>%left_join(final,by=c("ID"="name"))
coreTable<-coreTable%>%select(-cardId)
coreTable<-coreTable%>%filter(complete.cases(.))%>%
  mutate(undervalue=value-cost)

zooCore<-coreTable%>%filter(undervalue>0 & Cluster==3)

zooDeckAvailableNumber<-30
zooDeck<-zooCore%>%select(ID)%>%mutate(count=2)

zooDeckAvailableNumber<-zooDeckAvailableNumber-dim(zooDeck)[1]*2

i<-0
while(as.numeric(zooDeckAvailableNumber)>0){
  i=i+1
  zooPCA<-zooDeck%>%left_join(total.df,by="ID")
  center<-colMeans(zooPCA[,3:11])
  neighbors<-total.df%>%
    mutate(distance=sqrt((PCA1-center[1])^2+
            (PCA2-center[2])^2+(PCA3-center[3])^2+(PCA4-center[4])^2+(PCA5-center[5])^2)+(PCA6-center[6])^2+
             (PCA7-center[7])^2+(PCA8-center[8])^2+(PCA9-center[9])^2)%>%arrange(distance)%>%
    filter(!ID %in% zooDeck[,1])
  if(zooDeckAvailableNumber==1){
    newCard<-neighbors[i,]%>%mutate(count=1)%>%select(ID,count)
  }else{
    newCard<-neighbors[i,]%>%mutate(count=ifelse(rarity!="Legendary",2,1))%>%select(ID,count)
  }
  zooDeck<-rbind(zooDeck,newCard)
  zooDeckAvailableNumber<-zooDeckAvailableNumber-as.numeric(newCard[,2])
}
zooDeck<-read.csv("zooDeck.csv",header = TRUE,sep=",")

zooDeck[,2:3]%>%kable
ID count
Knife Juggler 2
Voidwalker 2
Doomguard 2
Flame Imp 2
Nerubian Egg 2
Dire Wolf Alpha 2
Power Overwhelming 2
Void Terror 2
Argent Squire 2
Dark Iron Dwarf 2
Sea Giant 2
Curse of Rafaam 2
Voodoo Doctor 2
Bane of Doom 2
Leeroy Jenkins 1
Harvest Golem 1

Real-time evaluation of zooDeck:

Generally speaking, our established deck performed pretty well in players vs. computer games.

Players vs. Computer Opponent Hand Results Comments 1 Warlock first W @ 8 mana 2 Warlock first W @ 9 mana 3 Mage second W @ 7 mana 4 Mage first W @ 7 mana 5 Hunter second W @ 7 mana 6 Hunter first W @ 7 mana 7 Warrior first W @ 7 mana 8 Warrior second W @ 8 mana 9 Shaman second W @ 9 mana 10 Shaman second L @ 8 mana 11 Druid second W @ 6 mana 12 Druid second W @ 6 mana 13 Priest first W @ 8 mana 14 Priest second W @ 8 mana 15 Rogue first W @ 7 mana 16 Rogue first W @ 8 mana 17 Paladin second W @ 6 mana 18 Paladin first L @ 8 mana

However, when the opponent is human (especially when they are top players that have better cards in hand), it is hard to tell the results..

Player vs. Player Opponent Hand Results Comments 1 W Opponent concede 2 L @ 6 mana 3 second W @ 7 mana 4 Hunter first L @ 7 mana 5 Paladin second L @ 10 mana 6 Paladin first L
7 Priest first L 8 Druid first L @ 7 mana 9 Hunter first L @ 10 mana 10 Warlock first L @ 5 mana 11 Hunter second L @ 6 mana